Flow in blood vessel with Catheter

plus rest of solution to HW3_number 4

Velocity profile in annular region.

DSolve[{μ/r D[ r v'[r],r] - G ==0, v[λ R] ==0, v[R] ==0},v[r],r]

v[r]/.%[[1]]

FullSimplify[%]

flow rate outside of tube.

Integrate[%3 r,{r, λ R, R}, Assumptions->{λ>0}]

velocity profile inside tube

DSolve[{μ/r D[ r v'[r],r] - G ==0, v'[0] ==0, v[λ R] ==0},v[r],r]

v[r]/.%[[1]]

inside flowrate

Integrate[% r,{r, 0 , λ R}, Assumptions->{λ>0}]

Ratio of outside to inside flows

%12/%6

FullSimplify[%,Assumptions->{0≤λ≤1}]

(((λ^4) Log[λ])/((((-1+(λ^2)))^2)-(-1+(λ^4)) Log[λ]))

No real hope here for an analytical solution Just find the root numerically.  

Note that R is completely out of the problem.

FindRoot[%==1,{λ,.5}]

LogLogPlot[%1,{λ,.1,.9}, AxesLabel->{"λ", "flowrate ratio"}]